Interlacing properties of zeros of multiple orthogonal polynomials

TL;DR

This paper investigates the interlacing properties of zeros of multiple orthogonal polynomials, providing new sufficient conditions based on recurrence relation coefficients, supported by examples.

Contribution

It introduces a novel sufficient condition for zero interlacing in multiple orthogonal polynomials using recurrence relation coefficients.

Findings

Sufficient condition for interlacing established

Recursion relations analyzed for zero distribution

Examples illustrating the main results provided

Abstract

It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main result gives a sufficient condition, based on the coefficients in the recurrence relations, for the interlacing of the zeros of neighboring multiple orthogonal polynomials. We give several examples illustrating our result.